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Random Variable
Motivation example In an opinion poll, we might decide to ask 50 people whether they agree
or disagree with a certain issue. If we record a “1” for agree and “0” for disagree, the sample
space for this experiment has 250 elements. If we define a variable X=number of 1s recorded
out of 50, we have captured the essence of the problem. Note that the sample space of X
is the set of integers {1, 2, . . . , 50} and is much easier to deal with than the original sample
space.
In defining the quantity X, we have defined a mapping (a function) from the original sample
space to a new sample space, usually a set of real numbers. In general, we have the following
definition.
Definition of Random Variable A random variable is a function from a sample space S into
the real numbers.
Example 1.4.2 (Random variables)
In some experiments random variables are implicitly used; some examples are these.
Experiment
Random variable
Toss two dice
X =sum of the numbers
Toss a coin 25 times
X =number of heads in 25 tosses
Apply different amounts of
fertilizer to corn plants
X =yield/acre
Suppose we have a sample space
S = {s1 , . . . , sn }
with a probability function P and we define a random variable X with range X = {x1 , . . . , xm }.
We can define a probability function PX on X in the following way. We will observe X = xi
if and only if the outcome of the random experiment is an sj ∈ S such that X(sj ) = xi .
Thus,
PX (X = xi ) = P ({sj ∈ S : X(sj ) = xi }).
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Remark:
1. FX is defined for all values of x, not just those in X = {0, 1, 2, 3}. Thus, for example,
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FX (2.5) = P (X ≤ 2.5) = P (X = 0, 1, 2) = .
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2. FX has jumps at the values of xi ∈ X and the size of the jump at xi is equal to
P (X = xi ).
3. FX = 0 for x < 0 since X cannot be negative, and FX (x) = 1 for x ≥ 3 since x is
certain to be less than or equal to such a value.
FX is right-continuous, namely, the function is continuous when a point is approached
from the right. The property of right-continuity is a consequence of the definition of the cdf.
In contrast, if we had defined FX (x) = PX (X < x), FX would then be left-continuous.
Theorem 1.5.3
The function FX (x) is a cdf if and only of the following three conditions hold:
a. limx→−∞ F (x) = 0 and limx→∞ F (x) = 1.
b. F (x) is a nondecreasing function of x.
c. F (x) is right-continuous; that is, for every number x0 , limx↓x0 F (x) = F (x0 ).
Example 1.5.4 (Tossing for a head) Suppose we do an experiment that consists of tossing
a coin until a head appears. Let p =probability of a head on any given toss, and define
X =number of tosses required to get a head. Then, for any x = 1, 2, . . .,
P (X = x) = (1 − p)x−1 p.
The cdf is
FX (x) = P (X ≤ x) =
x
X
P (X = i) =
i=1
x
X
(1 − p)i−1 p = 1 − (1 − p)x .
i=1
It is easy to show that if 0 < p < 1, then FX (x) satisfies the conditions of Theorem 1.5.3.
First,
lim FX (x) = 0
x→−∞
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